The symmetric sum of a function of variables is defined to be , where ranges over all permutations of .
More generally, a symmetric sum of variables is a sum that is unchanged by any permutation of its variables.
Any symmetric sum can be written as a polynomial of elementary symmetric sums.
A symmetric function of variables is a function that is unchanged by any permutation of its variables. The symmetric sum of a symmetric function therefore satisfies
Given variables and a symmetric function with , the notation is sometimes used to denote the sum of over all subsets of size in .
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